Experiment

Surface preparation and cell culture

To fabricate micro-patterned substrates, we start with 20 mL PDMS solution (Silgard, Dow Corning, Midland, Michigan) and pour it over diffraction gratings with slit separation of 3 µm and total surface area 25 mm × 25 mm. The PDMS films were left to polymerize for 48 h at room temperature, then peeled away from the diffraction gratings, and cured at 55 °C for 3 h. We use atomic force microscope (AFM) imaging to ensure that the pattern was successfully transferred from the diffraction grating to the PDMS surface [Figs. 1(a) and 1(b)]. The result is a series of periodic patterns (parallel lines with crests and troughs) with the distance between two adjacent lines d = 3 µm [Fig. 1(b)]. The surfaces were glued to glass slides using silicone glue and then dried for 48 h. Next, each surface was cleaned with sterile water and spin-coated with 3 mL of poly-D-lysine (PDL) (Sigma-Aldrich, St. Louis, MO, USA) solution of concentration 0.1 mg/mL. The spinning was performed for 10 min at 1000 RPM. Prior to cell culture, the surfaces have been sterilized using ultraviolet light for 30 min.

Figure 1.

(a) AFM topographical scan of the micro-patterned PDMS substrates. The arrows in the inset define the reference directions for the measured angular distributions. The growth angle θ is measured with respect to the direction specified by 0 rad. The growth angle is positive if the axon grows in the “upward” direction: 0 <0 <π, and negative if the axon grows “downwards”:− π<0 <0. Note that +π and − π rad specify the same direction. (b) Height profile (line scan) across a horizontal line. This height profile illustrates the repeating periodic pattern on the substrates with a periodicity d = 3 µm.

Cortical neurons have been obtained from rat embryos (day 18 embryos obtained from Tufts Medical School). The cortices have been incubated in 5 mL of trypsin at 37 °C for 20 min. To inhibit the trypsin, we have used 10 mL of soybean trypsin inhibitor (Life Technologies, Grand Island, NY, USA). Then the neuronal cells have been mechanically dissociated, centrifuged, and the supernatant was removed.^[Reference Spedden, White, Naumova, Kaplan and Staii^3] Next, the neurons have been resuspended in 20 mL of neurobasal medium (Life Technologies) enhanced with GlutaMAX, b27 (Life Technologies), and pen/strep. Finally, the neurons have been re-dispersed with a pipette, counted, and plated at a density of 5000 cells/cm².

Results and discussion

The substrates utilized in this study are PDL-coated PDMS surfaces, with periodic micro-patterns (parallel ridges separated by troughs, Fig. 1). Cortical neurons were plated on the substrates and axon growth was quantified at different time points after plating, in increments of 4 h: t = 8, 12, 16, 20, 24, and 28 h. At each time point, we have measured the total length of each axon as well as the axon distribution as a function of growth angle θ. Growth at θ =  ± π/2 rad is defined as axonal elongation in the direction of the micro-patterned surface lines (vertical direction in Fig. 1), while growth at θ = 0 and ±π rad represents growth in the direction perpendicular to the patterns (horizontal direction in Fig. 1).

Examples of axonal growth are shown in Fig. 2. To quantify the axonal angular distribution, we have divided each axon into segments of 20 µm in length using ImageJ software.

Figure 2.

Fluorescent images for neurons grown on PDMS substrates with periodic micro-patterns similar to the ones shown in Fig. 1. Neuronal cell bodies are shown by rounded features; axons are represented by the long processes outgrowing from the cell body; dendrites are represented by the short processes (for cortical neurons there is only one axon/cell body). (a) Image showing neuron growth 16 h after plating. The arrows in the inset define the reference directions for the measured angular distributions [same as in Fig. 1(a)]. (b) Image showing neuron growth 24 h after plating. Both (a) and (b) demonstrate a high degree of alignment between the direction of the axonal growth and the direction of the micro-patterns (θ =  ± π/2).

The dynamics of the growth cone is a stochastic process and is influenced by many types of phenomena, including fluctuations of the weak chemical gradients, the fundamental stochastic character of biochemical reactions taking place in the growth cone, polymerization/de-polymerization processes involved in actin and microtubule dynamics, etc. It follows that, quite generally the dynamics of the growth cone on micro-patterned surfaces is controlled both by a deterministic component (tendency to grow in certain preferred directions imparted by surface geometry), and a random deviation from these growth directions due to stochastic processes. In our previous work,^[Reference Spedden, Wiens, Demirel and Staii^10, Reference Rizzo, White, Spedden, Wiens, Kaplan, Atherton and Staii^11] we have used a general formalism based on the FP equation to describe the dynamics of the growth cones on nano-ppx or glass surfaces. A similar formalism has been used by other authors, for example, to describe the motion of human granulocyte cells under the influence of electric fields^[Reference Schienbein and Gruler^12] or the dynamics of the leading-edge lamellipodia of the growth cones.^[Reference Betz, Lim and Kas^13] Examples of normalized angular distributions for axons grown on micro-patterned PDMS surfaces measured at two different time points are shown in Figs. 3(a)–(d).

Figure 3.

Examples of normalized angular distributions for axonal growth at different time points: t = 16 h for (a) and (b) (N = 128 axons) and t = 24 h for (c) and (d) (N = 143 axons). The continuous curves represent fit of the data with Eq. (2). The distributions show peaks at two angles: θ = −π/2 and  + π/2, which correspond to axonal alignment along the micro-patterned lines (see Figs. 1 and 2). The fits to the data for the pairs of histograms: (a)–(d), respectively, give a single value for the ratio γ/D _θ at a given time point, as discussed in the text.

Using the angular coordinates defined in Fig. 1(a) for the micro-patterned PDMS surfaces, the FP equation is^[Reference Rizzo, White, Spedden, Wiens, Kaplan, Atherton and Staii^11, Reference Risken and Haken^14]:

(1)$$\displaystyle{{\partial p(\theta, t)} \over {\partial t}} = \displaystyle{\partial \over {\partial \theta}} [ - \gamma \cdot \cos \theta (t) \cdot p\theta (t)] + D_\theta \cdot \displaystyle{{\partial ^2p(\theta, t)} \over {\partial \theta ^2}},$$

where p(θ,t) is the probability distribution for growth angles and D _θ is an effective diffusion (cell motility) coefficient for the angular motion. As in Ref. [Reference Spedden, Wiens, Demirel and Staii10], we interpret the first term on the right-hand side of Eq. (1) as a “deterministic torque,” which tends to align the growth cone with the preferred direction determined by the surface geometry. There is no torque if the motion of the growth cone is parallel to the direction of micro-patterns (i.e., θ = π/2 or  −π/2 in Fig. 1). However, if the growth cone moves perpendicular to the patterns (θ = 0 or θ = π), the torque, and the cell–surface interaction that tends to align the axon, has a maximum value. The magnitude of γ quantifies the strength of the interaction between patterned PDMS surface and the growth cone. We note that Eq. (1) is a stochastic differential equation. In the absence of the deterministic torque γ, the angular motion of the growth cone is described by a simple random walk (diffusion) in angular space. The experimentally observed directional bias of axonal growth clearly indicates an additional interaction between the growth cones and the PDMS surfaces, which tends to align the axons with the micro-patterned features.

The angular distributions in Fig. 3 demonstrate aligned growth, with the axons preferentially growing along the micro-patterned lines (peaks at θ = π/2 and  −π/2, respectively). Furthermore, if we compare the peak heights for each pair of histograms shown in Figs. 3(a) versus 3(b) and 3(c) versus 3(d), we note that there is no polarized unidirectional growth, i.e., no statistically significant differences in the “upward” (θ = π/2) versus “downward” (θ = −π/2) motion of the growth cones for each PDMS surface. This conclusion holds at each time point considered in these experiments. This result is in contrast with the observed neuronal growth on a different type of directional surfaces (textured nano-ppx surfaces), where we have reported a supplementary unidirectional bias in addition to the preferred alignment along the surface asymmetry.^[Reference Spedden, Wiens, Demirel and Staii^10]

The stationary solution of Eq. (1) at each time point is given by^[Reference Risken and Haken^14]:

(2)$$p(0) = A \cdot {\rm Exp}\left[ {\displaystyle{\gamma \over {D_\theta}} \cdot \vert \sin (\theta ) \vert} \right],$$

where A is a normalization constant. We use Eq. (2) to fit the normalized experimental distributions at each time point considered in these experiments [fits to the data are represented by the continuous curves in Figs. 3(a)–(d)]. We note that at each time point, the experimentally determined angular distributions give only one value for the ratio γ/D _θ between the deterministic torque and the diffusion coefficient for the angular motion. For example, by fitting the distributions measured at t = 16 h with Eq. (2) we obtain γ/D _θ = (14.3 ± 0.8) [Fig. 3(a)] and γ/D _θ = (15.1 ± 0.9) [Fig. 3(b)]. The two values are the same within the estimated uncertainties. A similar fit for the distributions measured at t = 24 h gives: γ/D _θ = (5.6 ± 0.6) [Fig. 3(c)] and γ/D _θ = (5.1 ± 0.4) [Fig. 3(d)]. These results are in contrast with the previous work on neuronal growth on nano-ppx surfaces^[Reference Spedden, Wiens, Demirel and Staii^10] where the polarized growth leads to different values for the deterministic torques that control the axonal alignment parallel versus antiparallel with respect to the surface anisotropy.

To determine the effective angular diffusion coefficient D _θ, we use the joint fit method we have previously reported for growth on nano-ppx and glass surfaces^[Reference Spedden, Wiens, Demirel and Staii^10, Reference Rizzo, White, Spedden, Wiens, Kaplan, Atherton and Staii^11]: we perform a best fit for all the angular distributions and select that value of D _θ which maximizes the likelihood of measuring the experimentally observed data sets. Using this procedure, we perform a joint fit for all surfaces at a given time and obtain an effective value for the angular diffusion coefficient of D _θ = (86 ± 8) rad²/h. Remarkably, this value is similar with the value obtained for nano-ppx surfaces^[Reference Spedden, Wiens, Demirel and Staii^10]:$D_\theta = 92 \pm \displaystyle{5 \over 7}\;{\rm ra}{\rm d}^{\rm 2}/{\rm h}$, which indicates that axonal growth on the two types of directional surfaces (micro-patterned PDMS and nano-ppx) has similar diffusion properties. However, the axonal alignment properties and the axonal–surface interactions quantified by the deterministic torque γ are different, as we discuss below.

We use Eq. (2) to fit the experimental data and to quantify the variation with time of the ratio γ/D _θ between cell–surface coupling torque and the angular diffusion coefficient. Figure 4 shows this variation for the time series considered in these experiments t = 8, 12, 16, 20, 24, and 28 h. Giving that the angular diffusion coefficient has a constant value for all the considered time points, the data in Fig. 4 demonstrate that the magnitude of the deterministic torque γ decreases in time. This implies that the strength of the interaction that tends to align the axons to the patterned lines on PDMS decreases with time. We provide the following interpretation of this experimental result. At earlier times when axons start growing, the growth cone explores the surrounding environment by performing a random walk on the surface.^[Reference Wen and Zheng^1, Reference Lowery and van Vactor^2, Reference Beighley, Spedden, Sekeroglu, Atherton, Demirel and Staii^9–Reference Rizzo, White, Spedden, Wiens, Kaplan, Atherton and Staii^11] On patterned surfaces, cell–surface forces tend to align the motion of the growth cone along the preferred direction imparted by the surface features. These interactions are stronger at earlier times when the growth cone tends to move in random directions. Once aligned with the patterns, the axons continue to grow preferentially along the direction of the pattern and the corresponding torques that align the motion become less strong. That is, the higher the degree of alignment between the axon and the surface pattern, the less is the magnitude of the torque necessary to produce alignment. This is consistent with our data (Figs. 2 and 3) that shows a higher degree of alignment for later times, hence smaller values for γ. To our knowledge, this is a novel result not only for axonal growth but for the dynamics of any type of cell cultured on the directional surface.

Figure 4.

Variation in time of the ratio γ/D _θ between the deterministic torque and the diffusion coefficient for the angular motion.

It is interesting to compare the values of ratios γ/D _θ for growth on PDMS surfaces with the corresponding values measured for nano-ppx surfaces.^[Reference Spedden, Wiens, Demirel and Staii^10] On nano-ppx, we found these ratios in the range: 2 <γ/D _θ<8.^[Reference Spedden, Wiens, Demirel and Staii^10] Figure 4 shows that on PDMS, γ/D _θ varies between ~28 (t = 8 h) and ~4 (t = 28 h). Given that the angular diffusion coefficient D _θ is similar for the two types of surfaces, we conclude that the deterministic torques measured on the micro-patterns PDMS are significantly larger than the corresponding torques on asymmetric nano-ppx. This result is consistent with the higher degree of alignment on micro-patterned PDMS surfaces observed experimentally.

In the previous work,^[Reference Spedden, Wiens, Demirel and Staii^10] we have related the observed directional bias in axonal growth on nano-ppx surfaces to cellular contact guidance behavior, which leads to an enhanced cell–surface coupling with increasing surface anisotropy. Contact guidance is defined as the capacity of cells to change their motion in response to geometrical features (lines, periodic patterns, curved surfaces, etc.). This phenomenon has been reported for several types of cells, such as granulocytes, fibroblasts, tumor cells, and neurons. Our experiments show contact-guidance behavior for cortical neurons cultured on PDMS surfaces with micro-patterned parallel lines: axons grow preferentially in the direction of the lines. The periodicity of the patterns (3 µm) is comparable with the dimensions of the growth cones. Previous reports have demonstrated that the directional stimulation of cellular motion is enhanced by an increased density of anchored surface receptors.^[Reference Moore and Sheetz^15, Reference Frey, Tsai, Russell, Hanks and Wang^16] Inside the axons, the microtubules act as load-bearing structures: they are resistant to bending, and therefore produce a resistive force that opposes the bending of the growth cone. At the locations where the substrate curvature is large, the adhesion points for a growth cone bend around a steeper angle thus exerting larger forces and determining the creation of more focal adhesion points. Thus, the focal contacts on a growth cone wrapped around a high curvature structure on the PDMS surface (crest of the pattern) will experience higher traction forces than those contacting a smaller angle feature (slope of the patterns). This results in a larger value of the traction force along the features of higher curvature (crests) and consequently in a lower tendency for the axons to change the growth direction. Ultimately this leads to the alignment of axonal growth along the features of larger curvature (micro-pattern lines).

In this work, we have cultured neurons at a constant surface density (5000 cells/cm²), such that the effects of cell–cell interactions are expected to be roughly the same for all the different time points and PDMS surfaces. In the previous work,^[Reference Beighley, Spedden, Sekeroglu, Atherton, Demirel and Staii^9] we have explored the effects that a varying cell surface density has on the directional growth of neurons cultured on nano-ppx surfaces. We found that the directional bias imparted by surface geometry dominates the axonal alignment at low and medium cell densities (<10,000 cells/cm²) while the cell–cell interactions start to play an important role at high cell densities (above 10,000 cells/cm²). The effects of surface cell density and cell–cell interactions for neurons grown on PDMS surfaces will form the subject of a future study.